In a race of fluctuating fortunes and many quarrels on and off the track, the McLaren duo secured a comprehensive one-two after starting from the team's first front row lockout since 2012, Norris finally obeying team orders to hand his team-mate his first career win.
Piastri, 23, won by 2.141 seconds with seven-time champion Lewis Hamilton finishing third for Mercedes to claim his record 200th podium finish.
He survived a late collision with Red Bull's three-time champion and series leader Max Verstappen, who flew off, but recovered to finish fifth.
Charles Leclerc came home fourth and Ferrari team-mate Carlos Sainz sixth, sandwiching a grumpy Verstappen who was called to see the stewards to explain his collision with Hamilton.
Sergio Perez finished seventh for Red Bull, having started 16th on the grid, ahead of George Russell in the second Mercedes, who started 17th, and RB's Yuki Tsunoda. Lance Stroll was 10th for Aston Martin.
"It's very special," said Australian driver Piastri.
"I dreamt of this as a kid and if it was a bit complicated at the end, I did put myself in the right position at the start of the race.
"It's a hell of a lot of fun racing with McLaren. This is an incredible feeling."
Norris was first to congratulate his team-mate, after he had appeared to reject team orders and allow the Australian to pass in the closing stages.
"Well done, a good 1-2 and lots of good points for the team. Well deserved," he said.
Norris had made an uncertain start and he, Piastri and Verstappen were three abreast into Turn One where Piastri exited in the lead as the Dutchman ran wide and cut back into second place, gaining a clear advantage and pushing Norris down to third.
This prompted an exchange of messages before race engineer Gianpiero Lambiase told Verstappen to allow Norris to pass, a command that clearly irked him.
"So, you can just run people off the track?" barked the Dutchman.
'Do the right thing'
By lap 10, Piastri led Norris by 2.7 seconds with Verstappen third adrift by two seconds ahead of Hamilton and the two Ferraris, led by Leclerc.
Hamilton eventually reeled off a series of fastest laps to rise to third, but Verstappen on younger tyres reeled him in, waiting to pounce as the Briton endured a lurid slide out of Turn 12 before pitting again on lap 41 after fending off the Dutchman.
At the front, Piastri was in cruise mode ahead of Norris with Verstappen third, 11.5 seconds adrift. Hamilton rejoined fifth behind Sainz, but with Leclerc, on new mediums, on his tail.
Norris pitted again for mediums on lap 46, rejoining fourth ahead of Hamilton, followed by Piastri on 47, handing the lead to Verstappen with Norris up to second, but told to "re-establish the order at your convenience".
Verstappen made his second stop, for mediums, on lap 50, rejoining fifth behind Leclerc, but adrift of the Ferrari by 4.5.
In the lead, Norris was reminded of his team instructions and responsibilities as Piastri closed in.
"We know you'll do the right thing," said McLaren, but Norris, knowing he could reduce Verstappen's championship lead, stayed silent when told not to stress his tyres.
"Tell him to catch up, please," he said.
As McLaren's tensions boiled over, Verstappen lunged down the inside of Hamilton at Turn One on lap 63, but locked up and clipped the Mercedes. The collision sent him airborne briefly before he bounced clear and wide before rejoining in fifth.
McLaren then issued an ultimatum to Norris.
"There are five laps to go. The way to win a championship is not by yourself. It is with the team. You are going to need Oscar and you are going to need the team."
With three laps remaining, Norris slowed dramatically to gift Piastri the lead.
